Complex division, imag part

Percentage Accurate: 62.0% → 88.0%

Time: 11.2s

Alternatives: 9

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

Python

def code(a, b, c, d):
	return ((b * c) - (a * d)) / ((c * c) + (d * d))

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((b * c) - (a * d)) / ((c * c) + (d * d));
}

Python

def code(a, b, c, d):
	return ((b * c) - (a * d)) / ((c * c) + (d * d))

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 88.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.1 \cdot 10^{-71} \lor \neg \left(a \leq 1.3 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{1}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{c}{\mathsf{hypot}\left(d, c\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= a -7.1e-71) (not (<= a 1.3e-69)))
   (* (/ (fma b (/ c a) (- d)) (hypot d c)) (/ a (hypot d c)))
   (* b (* (/ 1.0 (hypot d c)) (/ c (hypot d c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((a <= -7.1e-71) || !(a <= 1.3e-69)) {
		tmp = (fma(b, (c / a), -d) / hypot(d, c)) * (a / hypot(d, c));
	} else {
		tmp = b * ((1.0 / hypot(d, c)) * (c / hypot(d, c)));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((a <= -7.1e-71) || !(a <= 1.3e-69))
		tmp = Float64(Float64(fma(b, Float64(c / a), Float64(-d)) / hypot(d, c)) * Float64(a / hypot(d, c)));
	else
		tmp = Float64(b * Float64(Float64(1.0 / hypot(d, c)) * Float64(c / hypot(d, c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[a, -7.1e-71], N[Not[LessEqual[a, 1.3e-69]], $MachinePrecision]], N[(N[(N[(b * N[(c / a), $MachinePrecision] + (-d)), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(1.0 / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(c / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.10000000000000008e-71 or 1.3000000000000001e-69 < a

   1. Initial program 49.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 49.6%

      \[\leadsto \frac{\color{blue}{a \cdot \left(\frac{b \cdot c}{a} - d\right)}}{c \cdot c + d \cdot d} \]
   4. Step-by-step derivation

      1. associate-/l* 48.4%

         \[\leadsto \frac{a \cdot \left(\color{blue}{b \cdot \frac{c}{a}} - d\right)}{c \cdot c + d \cdot d} \]

   5. Simplified 48.4%

      \[\leadsto \frac{\color{blue}{a \cdot \left(b \cdot \frac{c}{a} - d\right)}}{c \cdot c + d \cdot d} \]
   6. Step-by-step derivation

      1. *-commutative 48.4%

         \[\leadsto \frac{\color{blue}{\left(b \cdot \frac{c}{a} - d\right) \cdot a}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt 48.3%

         \[\leadsto \frac{\left(b \cdot \frac{c}{a} - d\right) \cdot a}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac 55.3%

         \[\leadsto \color{blue}{\frac{b \cdot \frac{c}{a} - d}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. fma-neg 55.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}} \]

      5. +-commutative 55.3%

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-define 55.3%

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\color{blue}{\mathsf{hypot}\left(d, c\right)}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}} \]

      7. +-commutative 55.3%

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}} \]

      8. hypot-define 90.5%

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\color{blue}{\mathsf{hypot}\left(d, c\right)}} \]

   7. Applied egg-rr 90.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}} \]

   if -7.10000000000000008e-71 < a < 1.3000000000000001e-69

   1. Initial program 72.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 65.1%

      \[\leadsto \color{blue}{\frac{b \cdot c}{{c}^{2} + {d}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* 66.7%

         \[\leadsto \color{blue}{b \cdot \frac{c}{{c}^{2} + {d}^{2}}} \]

   5. Simplified 66.7%

      \[\leadsto \color{blue}{b \cdot \frac{c}{{c}^{2} + {d}^{2}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 66.7%

         \[\leadsto b \cdot \frac{\color{blue}{1 \cdot c}}{{c}^{2} + {d}^{2}} \]

      2. pow2 66.7%

         \[\leadsto b \cdot \frac{1 \cdot c}{\color{blue}{c \cdot c} + {d}^{2}} \]

      3. pow2 66.7%

         \[\leadsto b \cdot \frac{1 \cdot c}{c \cdot c + \color{blue}{d \cdot d}} \]

      4. add-sqr-sqrt 66.7%

         \[\leadsto b \cdot \frac{1 \cdot c}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      5. times-frac 66.6%

         \[\leadsto b \cdot \color{blue}{\left(\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{c}{\sqrt{c \cdot c + d \cdot d}}\right)} \]

      6. +-commutative 66.6%

         \[\leadsto b \cdot \left(\frac{1}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}} \cdot \frac{c}{\sqrt{c \cdot c + d \cdot d}}\right) \]

      7. hypot-define 66.6%

         \[\leadsto b \cdot \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(d, c\right)}} \cdot \frac{c}{\sqrt{c \cdot c + d \cdot d}}\right) \]

      8. +-commutative 66.6%

         \[\leadsto b \cdot \left(\frac{1}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{c}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\right) \]

      9. hypot-define 87.4%

         \[\leadsto b \cdot \left(\frac{1}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{c}{\color{blue}{\mathsf{hypot}\left(d, c\right)}}\right) \]

   7. Applied egg-rr 87.4%

      \[\leadsto b \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{c}{\mathsf{hypot}\left(d, c\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.1 \cdot 10^{-71} \lor \neg \left(a \leq 1.3 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{1}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{c}{\mathsf{hypot}\left(d, c\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -9 \cdot 10^{-74}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - d \cdot \frac{a}{c}\right) \cdot \frac{1}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.5e+22)
   (/ (- b (* a (/ d c))) c)
   (if (<= c -9e-74)
     (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
     (if (<= c 4.2e-153)
       (/ (- (/ (* b c) d) a) d)
       (if (<= c 1.5e+75)
         (/ (fma b c (* d (- a))) (fma d d (* c c)))
         (* (- b (* d (/ a c))) (/ 1.0 c)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.5e+22) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -9e-74) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 4.2e-153) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 1.5e+75) {
		tmp = fma(b, c, (d * -a)) / fma(d, d, (c * c));
	} else {
		tmp = (b - (d * (a / c))) * (1.0 / c);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.5e+22)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= -9e-74)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 4.2e-153)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 1.5e+75)
		tmp = Float64(fma(b, c, Float64(d * Float64(-a))) / fma(d, d, Float64(c * c)));
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) * Float64(1.0 / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.5e+22], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -9e-74], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.2e-153], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.5e+75], N[(N[(b * c + N[(d * (-a)), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / c), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.5e22

   1. Initial program 36.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 71.8%

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. remove-double-neg 71.8%

         \[\leadsto \frac{\color{blue}{\left(-\left(-b\right)\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      2. mul-1-neg 71.8%

         \[\leadsto \frac{\left(-\color{blue}{-1 \cdot b}\right) + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      3. neg-mul-1 71.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. distribute-lft-in 71.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      5. mul-1-neg 71.8%

         \[\leadsto \frac{\color{blue}{-\left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      6. distribute-neg-in 71.8%

         \[\leadsto \frac{\color{blue}{\left(--1 \cdot b\right) + \left(-\frac{a \cdot d}{c}\right)}}{c} \]

      7. mul-1-neg 71.8%

         \[\leadsto \frac{\left(-\color{blue}{\left(-b\right)}\right) + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      8. remove-double-neg 71.8%

         \[\leadsto \frac{\color{blue}{b} + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      9. unsub-neg 71.8%

         \[\leadsto \frac{\color{blue}{b - \frac{a \cdot d}{c}}}{c} \]

      10. associate-/l* 80.8%

          \[\leadsto \frac{b - \color{blue}{a \cdot \frac{d}{c}}}{c} \]

   5. Simplified 80.8%

      \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{c}} \]

   if -1.5e22 < c < -8.9999999999999998e-74

   1. Initial program 99.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   if -8.9999999999999998e-74 < c < 4.20000000000000008e-153

   1. Initial program 65.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf 91.2%

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]

   if 4.20000000000000008e-153 < c < 1.5e75

   1. Initial program 80.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. fma-neg 80.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, c, -a \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. distribute-rgt-neg-out 80.5%

         \[\leadsto \frac{\mathsf{fma}\left(b, c, \color{blue}{a \cdot \left(-d\right)}\right)}{c \cdot c + d \cdot d} \]

      3. +-commutative 80.5%

         \[\leadsto \frac{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right)}{\color{blue}{d \cdot d + c \cdot c}} \]

      4. fma-define 80.5%

         \[\leadsto \frac{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right)}{\color{blue}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]

   3. Simplified 80.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
   4. Add Preprocessing

   if 1.5e75 < c

   1. Initial program 39.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 79.0%

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. remove-double-neg 79.0%

         \[\leadsto \frac{\color{blue}{\left(-\left(-b\right)\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      2. mul-1-neg 79.0%

         \[\leadsto \frac{\left(-\color{blue}{-1 \cdot b}\right) + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      3. neg-mul-1 79.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. distribute-lft-in 79.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      5. mul-1-neg 79.0%

         \[\leadsto \frac{\color{blue}{-\left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      6. distribute-neg-in 79.0%

         \[\leadsto \frac{\color{blue}{\left(--1 \cdot b\right) + \left(-\frac{a \cdot d}{c}\right)}}{c} \]

      7. mul-1-neg 79.0%

         \[\leadsto \frac{\left(-\color{blue}{\left(-b\right)}\right) + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      8. remove-double-neg 79.0%

         \[\leadsto \frac{\color{blue}{b} + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      9. unsub-neg 79.0%

         \[\leadsto \frac{\color{blue}{b - \frac{a \cdot d}{c}}}{c} \]

      10. associate-/l* 83.8%

          \[\leadsto \frac{b - \color{blue}{a \cdot \frac{d}{c}}}{c} \]

   5. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{c}} \]

   6. Taylor expanded in a around 0 79.0%

      \[\leadsto \frac{b - \color{blue}{\frac{a \cdot d}{c}}}{c} \]
   7. Step-by-step derivation

      1. *-commutative 79.0%

         \[\leadsto \frac{b - \frac{\color{blue}{d \cdot a}}{c}}{c} \]

      2. associate-*r/ 85.3%

         \[\leadsto \frac{b - \color{blue}{d \cdot \frac{a}{c}}}{c} \]

      3. *-commutative 85.3%

         \[\leadsto \frac{b - \color{blue}{\frac{a}{c} \cdot d}}{c} \]

      4. associate-/r/ 83.8%

         \[\leadsto \frac{b - \color{blue}{\frac{a}{\frac{c}{d}}}}{c} \]

   8. Simplified 83.8%

      \[\leadsto \frac{b - \color{blue}{\frac{a}{\frac{c}{d}}}}{c} \]
   9. Step-by-step derivation

      1. div-inv 83.7%

         \[\leadsto \color{blue}{\left(b - \frac{a}{\frac{c}{d}}\right) \cdot \frac{1}{c}} \]

      2. associate-/r/ 85.1%

         \[\leadsto \left(b - \color{blue}{\frac{a}{c} \cdot d}\right) \cdot \frac{1}{c} \]

   10. Applied egg-rr 85.1%

       \[\leadsto \color{blue}{\left(b - \frac{a}{c} \cdot d\right) \cdot \frac{1}{c}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -9 \cdot 10^{-74}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - d \cdot \frac{a}{c}\right) \cdot \frac{1}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -9 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{+73}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(b - d \cdot \frac{a}{c}\right) \cdot \frac{1}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= c -1.5e+22)
     (/ (- b (* a (/ d c))) c)
     (if (<= c -9e-72)
       t_0
       (if (<= c 1.12e-153)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 5.2e+73) t_0 (* (- b (* d (/ a c))) (/ 1.0 c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.5e+22) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -9e-72) {
		tmp = t_0;
	} else if (c <= 1.12e-153) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 5.2e+73) {
		tmp = t_0;
	} else {
		tmp = (b - (d * (a / c))) * (1.0 / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    if (c <= (-1.5d+22)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= (-9d-72)) then
        tmp = t_0
    else if (c <= 1.12d-153) then
        tmp = (((b * c) / d) - a) / d
    else if (c <= 5.2d+73) then
        tmp = t_0
    else
        tmp = (b - (d * (a / c))) * (1.0d0 / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.5e+22) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -9e-72) {
		tmp = t_0;
	} else if (c <= 1.12e-153) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 5.2e+73) {
		tmp = t_0;
	} else {
		tmp = (b - (d * (a / c))) * (1.0 / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.5e+22:
		tmp = (b - (a * (d / c))) / c
	elif c <= -9e-72:
		tmp = t_0
	elif c <= 1.12e-153:
		tmp = (((b * c) / d) - a) / d
	elif c <= 5.2e+73:
		tmp = t_0
	else:
		tmp = (b - (d * (a / c))) * (1.0 / c)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1.5e+22)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= -9e-72)
		tmp = t_0;
	elseif (c <= 1.12e-153)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 5.2e+73)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) * Float64(1.0 / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -1.5e+22)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= -9e-72)
		tmp = t_0;
	elseif (c <= 1.12e-153)
		tmp = (((b * c) / d) - a) / d;
	elseif (c <= 5.2e+73)
		tmp = t_0;
	else
		tmp = (b - (d * (a / c))) * (1.0 / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.5e+22], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -9e-72], t$95$0, If[LessEqual[c, 1.12e-153], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5.2e+73], t$95$0, N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.5e22

   1. Initial program 36.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 71.8%

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. remove-double-neg 71.8%

         \[\leadsto \frac{\color{blue}{\left(-\left(-b\right)\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      2. mul-1-neg 71.8%

         \[\leadsto \frac{\left(-\color{blue}{-1 \cdot b}\right) + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      3. neg-mul-1 71.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. distribute-lft-in 71.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      5. mul-1-neg 71.8%

         \[\leadsto \frac{\color{blue}{-\left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      6. distribute-neg-in 71.8%

         \[\leadsto \frac{\color{blue}{\left(--1 \cdot b\right) + \left(-\frac{a \cdot d}{c}\right)}}{c} \]

      7. mul-1-neg 71.8%

         \[\leadsto \frac{\left(-\color{blue}{\left(-b\right)}\right) + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      8. remove-double-neg 71.8%

         \[\leadsto \frac{\color{blue}{b} + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      9. unsub-neg 71.8%

         \[\leadsto \frac{\color{blue}{b - \frac{a \cdot d}{c}}}{c} \]

      10. associate-/l* 80.8%

          \[\leadsto \frac{b - \color{blue}{a \cdot \frac{d}{c}}}{c} \]

   5. Simplified 80.8%

      \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{c}} \]

   if -1.5e22 < c < -9e-72 or 1.12000000000000005e-153 < c < 5.2000000000000001e73

   1. Initial program 86.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   if -9e-72 < c < 1.12000000000000005e-153

   1. Initial program 65.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf 91.2%

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]

   if 5.2000000000000001e73 < c

   1. Initial program 39.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 79.0%

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. remove-double-neg 79.0%

         \[\leadsto \frac{\color{blue}{\left(-\left(-b\right)\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      2. mul-1-neg 79.0%

         \[\leadsto \frac{\left(-\color{blue}{-1 \cdot b}\right) + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      3. neg-mul-1 79.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. distribute-lft-in 79.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      5. mul-1-neg 79.0%

         \[\leadsto \frac{\color{blue}{-\left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      6. distribute-neg-in 79.0%

         \[\leadsto \frac{\color{blue}{\left(--1 \cdot b\right) + \left(-\frac{a \cdot d}{c}\right)}}{c} \]

      7. mul-1-neg 79.0%

         \[\leadsto \frac{\left(-\color{blue}{\left(-b\right)}\right) + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      8. remove-double-neg 79.0%

         \[\leadsto \frac{\color{blue}{b} + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      9. unsub-neg 79.0%

         \[\leadsto \frac{\color{blue}{b - \frac{a \cdot d}{c}}}{c} \]

      10. associate-/l* 83.8%

          \[\leadsto \frac{b - \color{blue}{a \cdot \frac{d}{c}}}{c} \]

   5. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{c}} \]

   6. Taylor expanded in a around 0 79.0%

      \[\leadsto \frac{b - \color{blue}{\frac{a \cdot d}{c}}}{c} \]
   7. Step-by-step derivation

      1. *-commutative 79.0%

         \[\leadsto \frac{b - \frac{\color{blue}{d \cdot a}}{c}}{c} \]

      2. associate-*r/ 85.3%

         \[\leadsto \frac{b - \color{blue}{d \cdot \frac{a}{c}}}{c} \]

      3. *-commutative 85.3%

         \[\leadsto \frac{b - \color{blue}{\frac{a}{c} \cdot d}}{c} \]

      4. associate-/r/ 83.8%

         \[\leadsto \frac{b - \color{blue}{\frac{a}{\frac{c}{d}}}}{c} \]

   8. Simplified 83.8%

      \[\leadsto \frac{b - \color{blue}{\frac{a}{\frac{c}{d}}}}{c} \]
   9. Step-by-step derivation

      1. div-inv 83.7%

         \[\leadsto \color{blue}{\left(b - \frac{a}{\frac{c}{d}}\right) \cdot \frac{1}{c}} \]

      2. associate-/r/ 85.1%

         \[\leadsto \left(b - \color{blue}{\frac{a}{c} \cdot d}\right) \cdot \frac{1}{c} \]

   10. Applied egg-rr 85.1%

       \[\leadsto \color{blue}{\left(b - \frac{a}{c} \cdot d\right) \cdot \frac{1}{c}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -9 \cdot 10^{-72}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\left(b - d \cdot \frac{a}{c}\right) \cdot \frac{1}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.9 \cdot 10^{-26} \lor \neg \left(c \leq 7.8 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.9e-26) (not (<= c 7.8e-102)))
   (/ (- b (* a (/ d c))) c)
   (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.9e-26) || !(c <= 7.8e-102)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-3.9d-26)) .or. (.not. (c <= 7.8d-102))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.9e-26) || !(c <= 7.8e-102)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.9e-26) or not (c <= 7.8e-102):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.9e-26) || !(c <= 7.8e-102))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3.9e-26) || ~((c <= 7.8e-102)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.9e-26], N[Not[LessEqual[c, 7.8e-102]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.89999999999999986e-26 or 7.79999999999999999e-102 < c

   1. Initial program 50.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 70.9%

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. remove-double-neg 70.9%

         \[\leadsto \frac{\color{blue}{\left(-\left(-b\right)\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      2. mul-1-neg 70.9%

         \[\leadsto \frac{\left(-\color{blue}{-1 \cdot b}\right) + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      3. neg-mul-1 70.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. distribute-lft-in 70.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      5. mul-1-neg 70.9%

         \[\leadsto \frac{\color{blue}{-\left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      6. distribute-neg-in 70.9%

         \[\leadsto \frac{\color{blue}{\left(--1 \cdot b\right) + \left(-\frac{a \cdot d}{c}\right)}}{c} \]

      7. mul-1-neg 70.9%

         \[\leadsto \frac{\left(-\color{blue}{\left(-b\right)}\right) + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      8. remove-double-neg 70.9%

         \[\leadsto \frac{\color{blue}{b} + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      9. unsub-neg 70.9%

         \[\leadsto \frac{\color{blue}{b - \frac{a \cdot d}{c}}}{c} \]

      10. associate-/l* 75.7%

          \[\leadsto \frac{b - \color{blue}{a \cdot \frac{d}{c}}}{c} \]

   5. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{c}} \]

   if -3.89999999999999986e-26 < c < 7.79999999999999999e-102

   1. Initial program 71.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 72.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. associate-*r/ 72.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 72.7%

         \[\leadsto \frac{\color{blue}{-a}}{d} \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{-a}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.9 \cdot 10^{-26} \lor \neg \left(c \leq 7.8 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{-65} \lor \neg \left(d \leq 1.05 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.15e-65) (not (<= d 1.05e+53)))
   (/ (- (* b (/ c d)) a) d)
   (/ (- b (/ a (/ c d))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.15e-65) || !(d <= 1.05e+53)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.15d-65)) .or. (.not. (d <= 1.05d+53))) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - (a / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.15e-65) || !(d <= 1.05e+53)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.15e-65) or not (d <= 1.05e+53):
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - (a / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.15e-65) || !(d <= 1.05e+53))
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.15e-65) || ~((d <= 1.05e+53)))
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - (a / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.15e-65], N[Not[LessEqual[d, 1.05e+53]], $MachinePrecision]], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.15e-65 or 1.0500000000000001e53 < d

   1. Initial program 46.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 39.0%

      \[\leadsto \frac{\color{blue}{a \cdot \left(\frac{b \cdot c}{a} - d\right)}}{c \cdot c + d \cdot d} \]
   4. Step-by-step derivation

      1. associate-/l* 35.4%

         \[\leadsto \frac{a \cdot \left(\color{blue}{b \cdot \frac{c}{a}} - d\right)}{c \cdot c + d \cdot d} \]

   5. Simplified 35.4%

      \[\leadsto \frac{\color{blue}{a \cdot \left(b \cdot \frac{c}{a} - d\right)}}{c \cdot c + d \cdot d} \]
   6. Step-by-step derivation

      1. *-commutative 35.4%

         \[\leadsto \frac{\color{blue}{\left(b \cdot \frac{c}{a} - d\right) \cdot a}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt 35.4%

         \[\leadsto \frac{\left(b \cdot \frac{c}{a} - d\right) \cdot a}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac 42.6%

         \[\leadsto \color{blue}{\frac{b \cdot \frac{c}{a} - d}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. fma-neg 42.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}} \]

      5. +-commutative 42.6%

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-define 42.6%

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\color{blue}{\mathsf{hypot}\left(d, c\right)}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}} \]

      7. +-commutative 42.6%

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}} \]

      8. hypot-define 75.0%

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\color{blue}{\mathsf{hypot}\left(d, c\right)}} \]

   7. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}} \]

   8. Taylor expanded in d around inf 70.7%

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   9. Step-by-step derivation

      1. +-commutative 70.7%

         \[\leadsto \frac{\color{blue}{\frac{b \cdot c}{d} + -1 \cdot a}}{d} \]

      2. mul-1-neg 70.7%

         \[\leadsto \frac{\frac{b \cdot c}{d} + \color{blue}{\left(-a\right)}}{d} \]

      3. sub-neg 70.7%

         \[\leadsto \frac{\color{blue}{\frac{b \cdot c}{d} - a}}{d} \]

      4. associate-/l* 76.9%

         \[\leadsto \frac{\color{blue}{b \cdot \frac{c}{d}} - a}{d} \]

   10. Simplified 76.9%

       \[\leadsto \color{blue}{\frac{b \cdot \frac{c}{d} - a}{d}} \]

   if -1.15e-65 < d < 1.0500000000000001e53

   1. Initial program 69.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 82.2%

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. remove-double-neg 82.2%

         \[\leadsto \frac{\color{blue}{\left(-\left(-b\right)\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      2. mul-1-neg 82.2%

         \[\leadsto \frac{\left(-\color{blue}{-1 \cdot b}\right) + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      3. neg-mul-1 82.2%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. distribute-lft-in 82.2%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      5. mul-1-neg 82.2%

         \[\leadsto \frac{\color{blue}{-\left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      6. distribute-neg-in 82.2%

         \[\leadsto \frac{\color{blue}{\left(--1 \cdot b\right) + \left(-\frac{a \cdot d}{c}\right)}}{c} \]

      7. mul-1-neg 82.2%

         \[\leadsto \frac{\left(-\color{blue}{\left(-b\right)}\right) + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      8. remove-double-neg 82.2%

         \[\leadsto \frac{\color{blue}{b} + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      9. unsub-neg 82.2%

         \[\leadsto \frac{\color{blue}{b - \frac{a \cdot d}{c}}}{c} \]

      10. associate-/l* 83.5%

          \[\leadsto \frac{b - \color{blue}{a \cdot \frac{d}{c}}}{c} \]

   5. Simplified 83.5%

      \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{c}} \]

   6. Taylor expanded in a around 0 82.2%

      \[\leadsto \frac{b - \color{blue}{\frac{a \cdot d}{c}}}{c} \]
   7. Step-by-step derivation

      1. *-commutative 82.2%

         \[\leadsto \frac{b - \frac{\color{blue}{d \cdot a}}{c}}{c} \]

      2. associate-*r/ 82.1%

         \[\leadsto \frac{b - \color{blue}{d \cdot \frac{a}{c}}}{c} \]

      3. *-commutative 82.1%

         \[\leadsto \frac{b - \color{blue}{\frac{a}{c} \cdot d}}{c} \]

      4. associate-/r/ 83.6%

         \[\leadsto \frac{b - \color{blue}{\frac{a}{\frac{c}{d}}}}{c} \]

   8. Simplified 83.6%

      \[\leadsto \frac{b - \color{blue}{\frac{a}{\frac{c}{d}}}}{c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{-65} \lor \neg \left(d \leq 1.05 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -8.2e-26)
   (/ (- b (* a (/ d c))) c)
   (if (<= c 7.8e-102) (/ a (- d)) (/ (- b (/ a (/ c d))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8.2e-26) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 7.8e-102) {
		tmp = a / -d;
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-8.2d-26)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= 7.8d-102) then
        tmp = a / -d
    else
        tmp = (b - (a / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8.2e-26) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 7.8e-102) {
		tmp = a / -d;
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -8.2e-26:
		tmp = (b - (a * (d / c))) / c
	elif c <= 7.8e-102:
		tmp = a / -d
	else:
		tmp = (b - (a / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -8.2e-26)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 7.8e-102)
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -8.2e-26)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= 7.8e-102)
		tmp = a / -d;
	else
		tmp = (b - (a / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -8.2e-26], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 7.8e-102], N[(a / (-d)), $MachinePrecision], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.1999999999999997e-26

   1. Initial program 48.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 71.6%

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. remove-double-neg 71.6%

         \[\leadsto \frac{\color{blue}{\left(-\left(-b\right)\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      2. mul-1-neg 71.6%

         \[\leadsto \frac{\left(-\color{blue}{-1 \cdot b}\right) + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      3. neg-mul-1 71.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. distribute-lft-in 71.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      5. mul-1-neg 71.6%

         \[\leadsto \frac{\color{blue}{-\left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      6. distribute-neg-in 71.6%

         \[\leadsto \frac{\color{blue}{\left(--1 \cdot b\right) + \left(-\frac{a \cdot d}{c}\right)}}{c} \]

      7. mul-1-neg 71.6%

         \[\leadsto \frac{\left(-\color{blue}{\left(-b\right)}\right) + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      8. remove-double-neg 71.6%

         \[\leadsto \frac{\color{blue}{b} + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      9. unsub-neg 71.6%

         \[\leadsto \frac{\color{blue}{b - \frac{a \cdot d}{c}}}{c} \]

      10. associate-/l* 78.9%

          \[\leadsto \frac{b - \color{blue}{a \cdot \frac{d}{c}}}{c} \]

   5. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{c}} \]

   if -8.1999999999999997e-26 < c < 7.79999999999999999e-102

   1. Initial program 71.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 72.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. associate-*r/ 72.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 72.7%

         \[\leadsto \frac{\color{blue}{-a}}{d} \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{-a}{d}} \]

   if 7.79999999999999999e-102 < c

   1. Initial program 52.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 70.4%

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. remove-double-neg 70.4%

         \[\leadsto \frac{\color{blue}{\left(-\left(-b\right)\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      2. mul-1-neg 70.4%

         \[\leadsto \frac{\left(-\color{blue}{-1 \cdot b}\right) + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      3. neg-mul-1 70.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. distribute-lft-in 70.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      5. mul-1-neg 70.4%

         \[\leadsto \frac{\color{blue}{-\left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      6. distribute-neg-in 70.4%

         \[\leadsto \frac{\color{blue}{\left(--1 \cdot b\right) + \left(-\frac{a \cdot d}{c}\right)}}{c} \]

      7. mul-1-neg 70.4%

         \[\leadsto \frac{\left(-\color{blue}{\left(-b\right)}\right) + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      8. remove-double-neg 70.4%

         \[\leadsto \frac{\color{blue}{b} + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      9. unsub-neg 70.4%

         \[\leadsto \frac{\color{blue}{b - \frac{a \cdot d}{c}}}{c} \]

      10. associate-/l* 73.5%

          \[\leadsto \frac{b - \color{blue}{a \cdot \frac{d}{c}}}{c} \]

   5. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{c}} \]

   6. Taylor expanded in a around 0 70.4%

      \[\leadsto \frac{b - \color{blue}{\frac{a \cdot d}{c}}}{c} \]
   7. Step-by-step derivation

      1. *-commutative 70.4%

         \[\leadsto \frac{b - \frac{\color{blue}{d \cdot a}}{c}}{c} \]

      2. associate-*r/ 74.4%

         \[\leadsto \frac{b - \color{blue}{d \cdot \frac{a}{c}}}{c} \]

      3. *-commutative 74.4%

         \[\leadsto \frac{b - \color{blue}{\frac{a}{c} \cdot d}}{c} \]

      4. associate-/r/ 73.5%

         \[\leadsto \frac{b - \color{blue}{\frac{a}{\frac{c}{d}}}}{c} \]

   8. Simplified 73.5%

      \[\leadsto \frac{b - \color{blue}{\frac{a}{\frac{c}{d}}}}{c} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{-65}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+52}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.15e-65)
   (/ (- (* b (/ c d)) a) d)
   (if (<= d 3.8e+52) (/ (- b (/ a (/ c d))) c) (/ (- (* c (/ b d)) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.15e-65) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= 3.8e+52) {
		tmp = (b - (a / (c / d))) / c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-1.15d-65)) then
        tmp = ((b * (c / d)) - a) / d
    else if (d <= 3.8d+52) then
        tmp = (b - (a / (c / d))) / c
    else
        tmp = ((c * (b / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.15e-65) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= 3.8e+52) {
		tmp = (b - (a / (c / d))) / c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.15e-65:
		tmp = ((b * (c / d)) - a) / d
	elif d <= 3.8e+52:
		tmp = (b - (a / (c / d))) / c
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.15e-65)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (d <= 3.8e+52)
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.15e-65)
		tmp = ((b * (c / d)) - a) / d;
	elseif (d <= 3.8e+52)
		tmp = (b - (a / (c / d))) / c;
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.15e-65], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 3.8e+52], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.15e-65

   1. Initial program 55.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf 46.4%

      \[\leadsto \frac{\color{blue}{a \cdot \left(\frac{b \cdot c}{a} - d\right)}}{c \cdot c + d \cdot d} \]
   4. Step-by-step derivation

      1. associate-/l* 42.3%

         \[\leadsto \frac{a \cdot \left(\color{blue}{b \cdot \frac{c}{a}} - d\right)}{c \cdot c + d \cdot d} \]

   5. Simplified 42.3%

      \[\leadsto \frac{\color{blue}{a \cdot \left(b \cdot \frac{c}{a} - d\right)}}{c \cdot c + d \cdot d} \]
   6. Step-by-step derivation

      1. *-commutative 42.3%

         \[\leadsto \frac{\color{blue}{\left(b \cdot \frac{c}{a} - d\right) \cdot a}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt 42.3%

         \[\leadsto \frac{\left(b \cdot \frac{c}{a} - d\right) \cdot a}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac 46.9%

         \[\leadsto \color{blue}{\frac{b \cdot \frac{c}{a} - d}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. fma-neg 46.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}} \]

      5. +-commutative 46.9%

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-define 46.9%

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\color{blue}{\mathsf{hypot}\left(d, c\right)}} \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}} \]

      7. +-commutative 46.9%

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}} \]

      8. hypot-define 75.2%

         \[\leadsto \frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\color{blue}{\mathsf{hypot}\left(d, c\right)}} \]

   7. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}} \]

   8. Taylor expanded in d around inf 72.9%

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   9. Step-by-step derivation

      1. +-commutative 72.9%

         \[\leadsto \frac{\color{blue}{\frac{b \cdot c}{d} + -1 \cdot a}}{d} \]

      2. mul-1-neg 72.9%

         \[\leadsto \frac{\frac{b \cdot c}{d} + \color{blue}{\left(-a\right)}}{d} \]

      3. sub-neg 72.9%

         \[\leadsto \frac{\color{blue}{\frac{b \cdot c}{d} - a}}{d} \]

      4. associate-/l* 76.4%

         \[\leadsto \frac{\color{blue}{b \cdot \frac{c}{d}} - a}{d} \]

   10. Simplified 76.4%

       \[\leadsto \color{blue}{\frac{b \cdot \frac{c}{d} - a}{d}} \]

   if -1.15e-65 < d < 3.8e52

   1. Initial program 69.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 82.2%

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. remove-double-neg 82.2%

         \[\leadsto \frac{\color{blue}{\left(-\left(-b\right)\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      2. mul-1-neg 82.2%

         \[\leadsto \frac{\left(-\color{blue}{-1 \cdot b}\right) + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      3. neg-mul-1 82.2%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b\right)} + -1 \cdot \frac{a \cdot d}{c}}{c} \]

      4. distribute-lft-in 82.2%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      5. mul-1-neg 82.2%

         \[\leadsto \frac{\color{blue}{-\left(-1 \cdot b + \frac{a \cdot d}{c}\right)}}{c} \]

      6. distribute-neg-in 82.2%

         \[\leadsto \frac{\color{blue}{\left(--1 \cdot b\right) + \left(-\frac{a \cdot d}{c}\right)}}{c} \]

      7. mul-1-neg 82.2%

         \[\leadsto \frac{\left(-\color{blue}{\left(-b\right)}\right) + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      8. remove-double-neg 82.2%

         \[\leadsto \frac{\color{blue}{b} + \left(-\frac{a \cdot d}{c}\right)}{c} \]

      9. unsub-neg 82.2%

         \[\leadsto \frac{\color{blue}{b - \frac{a \cdot d}{c}}}{c} \]

      10. associate-/l* 83.5%

          \[\leadsto \frac{b - \color{blue}{a \cdot \frac{d}{c}}}{c} \]

   5. Simplified 83.5%

      \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{c}} \]

   6. Taylor expanded in a around 0 82.2%

      \[\leadsto \frac{b - \color{blue}{\frac{a \cdot d}{c}}}{c} \]
   7. Step-by-step derivation

      1. *-commutative 82.2%

         \[\leadsto \frac{b - \frac{\color{blue}{d \cdot a}}{c}}{c} \]

      2. associate-*r/ 82.1%

         \[\leadsto \frac{b - \color{blue}{d \cdot \frac{a}{c}}}{c} \]

      3. *-commutative 82.1%

         \[\leadsto \frac{b - \color{blue}{\frac{a}{c} \cdot d}}{c} \]

      4. associate-/r/ 83.6%

         \[\leadsto \frac{b - \color{blue}{\frac{a}{\frac{c}{d}}}}{c} \]

   8. Simplified 83.6%

      \[\leadsto \frac{b - \color{blue}{\frac{a}{\frac{c}{d}}}}{c} \]

   if 3.8e52 < d

   1. Initial program 31.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 57.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 57.5%

         \[\leadsto \color{blue}{\frac{b \cdot c}{{d}^{2}} + -1 \cdot \frac{a}{d}} \]

      2. mul-1-neg 57.5%

         \[\leadsto \frac{b \cdot c}{{d}^{2}} + \color{blue}{\left(-\frac{a}{d}\right)} \]

      3. unsub-neg 57.5%

         \[\leadsto \color{blue}{\frac{b \cdot c}{{d}^{2}} - \frac{a}{d}} \]

      4. unpow2 57.5%

         \[\leadsto \frac{b \cdot c}{\color{blue}{d \cdot d}} - \frac{a}{d} \]

      5. associate-/r* 67.3%

         \[\leadsto \color{blue}{\frac{\frac{b \cdot c}{d}}{d}} - \frac{a}{d} \]

      6. div-sub 67.3%

         \[\leadsto \color{blue}{\frac{\frac{b \cdot c}{d} - a}{d}} \]

      7. *-commutative 67.3%

         \[\leadsto \frac{\frac{\color{blue}{c \cdot b}}{d} - a}{d} \]

      8. associate-/l* 77.9%

         \[\leadsto \frac{\color{blue}{c \cdot \frac{b}{d}} - a}{d} \]

   5. Simplified 77.9%

      \[\leadsto \color{blue}{\frac{c \cdot \frac{b}{d} - a}{d}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{-65}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+52}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9.5 \cdot 10^{-35} \lor \neg \left(c \leq 0.0031\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -9.5e-35) (not (<= c 0.0031))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -9.5e-35) || !(c <= 0.0031)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-9.5d-35)) .or. (.not. (c <= 0.0031d0))) then
        tmp = b / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -9.5e-35) || !(c <= 0.0031)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -9.5e-35) or not (c <= 0.0031):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -9.5e-35) || !(c <= 0.0031))
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -9.5e-35) || ~((c <= 0.0031)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -9.5e-35], N[Not[LessEqual[c, 0.0031]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -9.5000000000000003e-35 or 0.00309999999999999989 < c

   1. Initial program 48.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 65.4%

      \[\leadsto \color{blue}{\frac{b}{c}} \]

   if -9.5000000000000003e-35 < c < 0.00309999999999999989

   1. Initial program 72.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 68.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. associate-*r/ 68.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 68.3%

         \[\leadsto \frac{\color{blue}{-a}}{d} \]

   5. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{-a}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.5 \cdot 10^{-35} \lor \neg \left(c \leq 0.0031\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{b}{c} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (/ b c))

C

double code(double a, double b, double c, double d) {
	return b / c;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = b / c
end function

Java

public static double code(double a, double b, double c, double d) {
	return b / c;
}

Python

def code(a, b, c, d):
	return b / c

Julia

function code(a, b, c, d)
	return Float64(b / c)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = b / c;
end

Wolfram

code[a_, b_, c_, d_] := N[(b / c), $MachinePrecision]

Derivation

1. Initial program 58.4%

   \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
2. Add Preprocessing

3. Taylor expanded in c around inf 44.8%

   \[\leadsto \color{blue}{\frac{b}{c}} \]

4. Final simplification 44.8%

   \[\leadsto \frac{b}{c} \]
5. Add Preprocessing

Developer target: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (- b (* a (/ d c))) (+ c (* d (/ d c))))
   (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (fabs(d) < fabs(c)) {
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (abs(d) < abs(c)) then
        tmp = (b - (a * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (-a + (b * (c / d))) / (d + (c * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (abs(d) < abs(c))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(Float64(-a) + Float64(b * Float64(c / d))) / Float64(d + Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (abs(d) < abs(c))
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Less[N[Abs[d], $MachinePrecision], N[Abs[c], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) + N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024050 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))